# 英国补考|现代代数代写Modern Algebra代考|MAT 523

#### Doug I. Jones

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## 英国补考|现代代数代写Modern Algebra代考|Clock Arithmetic

On a 24-hour clock, let the cyclic group $\langle a\rangle$ of order 24 represent hours. Then each of 24 numerals of the dial serves as representatives of a coset of hours. Here we use the fact that a 10 hour journey starting at 20 hrs (8 o’clock) ends at 30 hrs, i.e., $10+20=30 \equiv 6$ o’clock (the following day). In shifting from a 24hour clock to a 12 hour clock, we take the 24 hours modulo the normal subgroup $H=\left{1, a^{12}\right}$. Then the quotient group $\langle a\rangle / H$ is a group of 12 cosets or a group of representatives: $\left{a, a^{2}, a^{3}, a^{11}, a^{12}\right}$, where $(a)=\left{a, a^{13}\right},\left(a^{2}\right)=\left{a^{2}, a^{14}\right},\left(a^{3}\right)=$ $\left{a^{3}, a^{15}\right},\left(a^{11}\right)=\left{a^{11}, a^{23}\right}$ and $\left(a^{12}\right)=\left{a^{12}, a^{24}\right}$. A seven hour journey starting at $9 \mathrm{AM}(9 \mathrm{PM})$ ends at $4 \mathrm{PM}(4 \mathrm{AM})$. On the 12-hour clock we do not distinguish between the congruent elements in the same coset ${4 \mathrm{AM}, 4 \mathrm{PM}}$.

A Note on Symmetry Group Group theory is an ideal tool to study symmetries. In this note we call any subset of $\mathbf{R}^{2}$ or $\mathbf{R}^{3}$ an object. We study orthogonal maps from $\mathbf{R}^{2}$ to $\mathbf{R}^{2}$ or from $\mathbf{R}^{3}$ to $\mathbf{R}^{3}$ and these are rotations or reflections or rotationreflections. As orthogonal maps are only those linear maps which keep lengths and angles invariant, they do not include translations.

Let $X$ be an object in $\mathbf{R}^{2}$ or $\mathbf{R}^{3}$. Then the set $S(X)$ of all orthogonal maps $g$ with $g(X)=X$ is a group with respect to the usual composition of maps. This group is called the symmetry group of $X$. The elements of $S(X)$ are called symmetry operations on $X$. Let $S_{2}(X)$ (or $S_{3}(X)$ ) correspond to the symmetric group of $X$ in $\mathbf{R}^{2}\left(\right.$ or $\left.\mathbf{R}^{3}\right)$

Examples (a) If $X=$ regular $n$-gon in $\mathbf{R}^{2}$ with center at $(0,0)$, then $S_{2}(X)=D_{n}$, the dihedral group with $2 n$ elements and for $S_{3}(X)$, we also get the reflection about the $x y$-plane and its compositions with all $g \in S_{2}(X)$, i.e., $S_{3}(X)=D_{n} \times \mathbf{Z}{2}$. (b) If $X=$ regular tetrahedron with center at $(0,0,0)$, then $S{3}(X) \cong S_{4}$.
(c) If $X=$ cube, then $S_{3}(X)=S_{4} \times \mathbf{Z}{2}$. (d) If $X=$ the letter $M$, then $S{2}(X) \cong \mathbf{Z}{2}$ and $S{3}(X) \cong \mathbf{Z}{2} \times \mathbf{Z}{2}$.
(e) If $X=$ a circle, then $S_{2}(X)=O_{2}(\mathbf{R})$, the whole orthogonal group of all orthogonal maps from $\mathbf{R}^{2}$ to $\mathbf{R}^{2}$.

## 英国补考|现代代数代写Modern Algebra代考|Free Abelian Groups and Structure Theorem

If we look at the dihedral group $D_{n}$ (see Ex. 19 of SE-III) we see that $D_{n}$ has two generators $r$ and $s$ satisfying some relations other than the associative property. But we now consider groups which have a set of generators satisfying no relations other than associativity which is implied by the group axioms. Such groups are called free groups. In this section we study free abelian groups and prove the ‘Fundamental Theorem of Finitely Generated Abelian Groups’ which is a structure theorem. This theorem gives notions of ‘Betti numbers’ and ‘invariant factors’. We also introduce the concept of ‘homology and cohomology groups’ which are very important in homological algebra and algebraic topology.

A free abelian groups is a direct sum of copies of additive abelian group of integers $\mathbf{Z}$. It has some properties similar to vector spaces. Every free abelian group has a basis and its rank is defined as the cardinality of a basis. The rank determines the groups up to isomorphisms and the elements of such a group can be written as finite formal sums of the basis elements.

The concept of free abelian groups is very important in mathematics. It has wide applications in homology theory. Algebraic topology is also used to prove some interesting properties of free abelian groups [see Rotman (1988)].
We consider in this section an additive abelian group $G$.
Definition 2.8.1 Let $G$ be an additive abelian group and $\left{G_{i}\right}_{i \in I}$ be a family of subgroups of $G$. Then $G$ is said to be generated by $\left{G_{i}\right}$ iff every element $x \in G$ can be expressed as
$x=x_{i_{1}}+\cdots+x_{i n}, \quad$ where the additive indices $i_{t}$ are distinct.
We sometimes use the following notation:
$x=\sum_{i \in I} x_{i}, \quad$ where we take $x_{i}=0$, if $i$ is not one of the indices $i_{1}, i_{2}, \ldots, i_{n} .$
If the subgroups $\left{G_{i}\right}$ generate $G$, we write
$$G=\sum_{i \in I} G_{i}, \quad \text { in general, } \quad \text { and } \quad G=\sum_{i=1}^{n} G_{i}, \quad \text { when } I={1,2, \ldots, n} .$$

# 现代代数代考

## 英国补考|现代代数代写Modern Algebra代考|Clock Arithmetic

(c) 如果 $X=$ 立方体，然后 $S_{3}(X)=S_{4} \times \mathbf{Z} 2$. (d) 如果 $X=$ 信 $M$ ，然后 $S 2(X) \cong \mathbf{Z} 2$ 和 $S 3(X) \cong \mathbf{Z} 2 \times \mathbf{Z} 2$.
(e) 如果 $X=$ 个圆圈，然后 $S_{2}(X)=O_{2}(\mathbf{R})$, 所有正交映射的整个正交组来自 $\mathbf{R}^{2}$ 至 $\mathbf{R}^{2}$.

## 英国补考|现代代数代写Modern Algebra代考|Free Abelian Groups and Structure Theorem

$x=x_{i_{1}}+\cdots+x_{i n}$, 其中附加指数 $i_{t}$ 是不同的。

$x=\sum_{i \in I} x_{i}, \quad$ 我们在哪里 $x_{i}=0$ ， 如果 $i$ 不是指数之一 $i_{1}, i_{2}, \ldots, i_{n}$.

$G=\sum_{i \in I} G_{i}, \quad$ in general, $\quad$ and $\quad G=\sum_{i=1}^{n} G_{i}, \quad$ when $I=1,2, \ldots, n$

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